Proving this seems difficult at first, but Cantor’s reasoning was so simple and original that it is worth understanding it. His is a ‘proof by contradiction’. Here we start with a statement that is opposite of what we want to prove and then show that this leads to an inconsistency.
The essence of Cantor’s proof is that we assume that the decimals (irrational & rational numbers both put together) are as numerous as the natural numbers ( i.e. integers, and we have shown above that the rational numbers have the same infinity א0, as the integers themselves) then it should be possible to label them one by one.
Let us accordingly number the list a1, a2, a3, a4 and so on :-